Center of Mass
The center of mass is a point in an object in which its mass is concentrated. Each object (such as a ball, glass, ruler, or pigeon) can be thought of as a collection of mass points. When an object is supported at its center of mass, the mass is at equilibrium and (1) is balanced and (2) can rotate uniformly. The center of mass for a sphere is located at its center. If the sphere is supported at its center, the sphere would be able to be balanced and able to spin uniformly. However, the center of mass of objects can be counter intuitive to its geometric center. For example, a triangle’s center of mass is closer to its base. A race car is engineered to have more weight towards the bottom of the car so drivers can handle them during high speed turns. An object’s center of mass will always be towards its more massive end. Center of Gravity According to Newton’s law of universal gravitation, the earth exerts a force upon objects on earth called gravity. The resultant force from an object’s mass and gravitational force is called its weight. This resultant goes through an object’s center of gravity. In a system containing the universal gravitation field, the weight of the object is concentrated at the center of gravity. The term center of gravity is used interchangeably with center of mass as long as the object experiences only earth’s gravitational force. When an object is thrown in the air, its center of gravity would follow a parabolic path. The figure to the right shows a portion of a wrench at it is tossed in the air. The center of gravity of a wrench follows a path, even though other parts of the wrench do not. The center of gravity also moves equal distances in equal time intervals (because no force/acceleration is acting upon the wrench). Locating the Center of Mass Irregular Shaped Objects Center of Mass Outside of Physical Structure of Object The center of mass of an object may exist where there is no mass. A donut’s center of mass is at its center. This holds true for a hollow sphere such as a soccer ball. Even though the object’s mass is not concentrated at its center of gravity, when the object is tossed in the air, its center of gravity would follow a parabolic path. Other objects that have their center of gravity outside their physical structures include an empty pan or cup, a chair, or a boomerang. Toppling An object will topple once its plumb line falls outside of its base of support. The figure to the right shows a block being toppled over once its plumb line falls outside the base of the box. Applications Leaning Tower of Pisa The Leaning Tower of Pisa is able to stand tilted without toppling over because the plumb line drawn from its center of mass is within the base of support. Interestingly enough, the tower's lean is a result of the tower's weak foundation from unstable subsoil. Historians, mathmaticians, and engineers from around the world throughout history have attempted to stabilize the tower and keep it from toppling. As of now, the Leaning Tower is standing in stability. Animal Tails The horizontal distance to which an animal body can extend depends on how far it can keep its center of gravity within its base of support. A monkey can reach farther by extending its tail, keeping its center of gravity within the support of its feet. By extending its tail, it can shift its center of gravity to maintain balance and stability. Dinosaurs such as the Brachiosaurus had massive tails to help them keep their center of gravity above their feet so they can extend their heads. Balancing Birds The masses of the balancing birds are strategically distributed on their wings so that the masses of the wings equal to the masses of their tails and bodies. The even distribution of mass at their beaks makes the beaks the center of masses. The birds will balance and rotate on any solid object until its torque restores it to its stationary position. Soda Can Trick It is possible to utilize center of mass to make a soda can balance on an edge. When there is only one-third of the soda left in the can, the soda can be balanced upon one edge because its center of mass would fall within the base of support when tilted. Trivia * The center of mass of a human body is at the pelvis area. * The center of mass of the solar system (when all planets are aligned collinearly) is about 2 solar radii from the sun’s center. * The center of mass of a rectangle is at the intersection of the two diagonals. * Alexander Calder, the inventor of mobiles, is famous for his sculptures that allow gusts of wind to arrange their elements. The structures of his mobiles are rearranged but the center of gravity always falls within the base of support (i.e. the pivot point). * Archimedes introduced the concept of center of gravity. He demonstrated that a single point on a lever is exerted the same amount of torque as weights resting at various points of the lever. He also developed methods to find centers of masses of regular shapes such as a triangle and hemisphere. Brainteasers # There are three trucks parked on a hill. The center of mass of each truck is marked with an “x”. Out of the three, which truck(s) would topple over? # A bottle rack that seems to defy common sense is shown in the figure. Where is the center of gravity of the rack and bottle? Solutions # Only the first truck will topple over. Draw a plumb line from the center of mass of each truck to the ground. Only the first truck’s plumb line falls out of its base of support (i.e. the ground directly below the truck). # The center of gravity is at point of the bottle directly above the base of support of the rack on the table. The center of gravity is over where the rack stands. It does not tip over because there is a a support beneath it. See also *Centroid *Weight Distribution *Torque *Alexander Calder *Center of Pressure *Rotation * References #“Center of mass.” Wikipedia, The Free Encyclopedia. 25 May 2006, 01:45 UTC. Wikimedia Foundation, Inc. 4 Jun 2006 <http://en.wikipedia.org/w/index.php?title=Center_of_mass&oldid=54995697>. #“Fall of the Leaning Tower.” NOVA Online. PBS. 4 Jun 2006 <http://www.pbs.org/wgbh/nova/pisa/>. #Gewirtz, Herman, and Jonathan S. Wolf. Barron’s How to Prepare for the Sat II Physics. 8th ed. New York: Barron's Educational Series, Inc., 2004. #Gibson, T.L.. “Physics 1303.” Physics Department. 01 Oct 20001. Texas Tech University. 4 Jun 2006 <http://www.phys.ttu.edu/~ritlg/courses/p1303/quiz8_js.html>. #Hewitt, Paul. Conceptual Physics. 3rd Tchr edition. New York: Addison-Wesley Professional, 1999. External links *Center of Mass - A visual and interactive site on center of mass and its applications. *Balancing Bird - Purchase a balancing bird at Arbor Scientific, a site that offers physics, chemistry, and physical science education supplies. *[http://www.phys.ttu.edu/~ritlg/courses/p1303/quiz8_js.html Center of Mass in Conceptual Physics] – Quiz yourself on center of mass. *Sparknotes: SAT Physics - Find equations involved in determining an object’s center of mass. *Continuous Distribution of Mass – Find the center of mass on a body of continuous distribution of mass. *Center of Mass Experiment - Complete a center of mass experiment with just a meter stick, clay, and masking tape. *Geometric Shapes – Center of masses of geometric shapes.